Root Loci For Positive-feedback Systems
Root loci for positive-feedback systems
Other rules for constructing the root-locus plot remain the same. We shall now apply the modified rules to construct the root-locus plot.
1. Plot the open-loop poles (s = -1 j, s = -1 - j, s = -3) and zero (s = -2) in the complex plane. As K is increased from 0 to ∞, the closed-loop poles start at the open loop poles and terminate at the open-loop zeros (finite or infinite), just as in the case of negative-feedback systems.
2. Determine the root loci on the real axis. Root loci exist on the real axis between – 2 and ∞ and between -3 and - ∞.
3. Determine the asymptotes of the root loci. For the present system,
This simply means that asymptotes are on the real axis.
4. Determine the breakaway and break-in points. Since the characteristic equation is
Point s = -0.8 is on the root locus. Since this point lies between two zeros (a finite zero and an infinite zero), it is an actual break-in point. Points s = -2.35 ± j0.77 do not satisfy the angle condition and, therefore, they are neither breakaway nor break-in points.
5. Find the angle of departure of the root locus from a complex pole. For the complex pole at s = -1 j, the angle of departure θ is
or θ = -720
(The angle of departure from the complex pole at s = -1 - j is 72°.)
6. Choose a test point in the broad neighborhood of the jω axis and the origin and apply the angle condition. Locate a sufficient number of points that satisfy the angle condition.
Figure 2 shows the root loci for the given positive-feedback system. The root loci are shown with dashed lines and curve.